GSEB Class 11 Statistics Solutions Chapter 4 Measures of Dispersion Solution Exercise 4.1

GSEB Solutions

GSEB Solutions Class 11 Statistics Chapter 4 Measures Of Dispersion Ex 4.1

Gujarat Board Textbook Solutions Class 11 Statistics Chapter 4 Measures Of Dispersion Ex 4.1

Question 1. The following data refer to the heights in cms. of 10 students of a class. Find range and coefficient of range of height of the students.
162, 145, 170, 181, 167,
151, 175, 185, 169, 156
Answer:
From the provided data, we identify the maximum height as \(x_H = 185\) cm and the minimum height as \(x_L = 145\) cm.

To calculate the range of height:
Range \(R = x_H - x_L\)
\(R = 185 - 145\)
\(R = 40\) cm

To calculate the coefficient of range:
Coefficient of range \( = \frac{x_H - x_L}{x_H + x_L}\)
\( = \frac{185 - 145}{185 + 145}\)
\( = \frac{40}{330}\)
\( = 0.1212\) (approximately)
In simple words: To find the range, subtract the smallest value from the largest value. The coefficient of range further expresses this spread relative to the sum of the smallest and largest values.

🎯 Exam Tip: Accurately identifying the highest and lowest values in a dataset is fundamental for correctly calculating both the range and its coefficient.

Question 2. A bus company has 77 buses for travelling in the city. The information of number of passengers in bus on a particular day at a particular time is given below. Find the range and coefficient of range of number of passengers.

No. of passengers	2	7	10	18	25	30	37
No. of buses	1	4	11	17	23	16	5

Answer:
Based on the given data, the maximum number of passengers observed in a bus is \(x_H = 37\), and the minimum number of passengers is \(x_L = 2\).

To determine the range of passenger numbers:
Range \(R = x_H - x_L\)
\(R = 37 - 2\)
\(R = 35\) passengers

To calculate the coefficient of range:
Coefficient of range \( = \frac{R}{x_H + x_L}\)
\( = \frac{35}{37 + 2}\)
\( = \frac{35}{39}\)
\( = 0.8974\) (approximately)
In simple words: The range tells us the total spread of passenger numbers, while the coefficient of range shows how large this spread is relative to the total number of passengers at the extremes.

🎯 Exam Tip: For discrete data, ensure you identify the absolute highest and lowest values from the 'No. of passengers' row, irrespective of their frequencies, to compute the range accurately.

Question 3. Using the following frequency distribution of marks of students of a school, find range and relative range of the marks:

Marks	20-30	30-40	40-50	50-60	60-70	70-80
No. of students	8	20	25	60	45	10

Answer:
For this frequency distribution, the upper limit of the last class is \(x_H = 80\) marks, and the lower limit of the initial class is \(x_L = 20\) marks.

To find the range of marks:
Range \(R = x_H - x_L\)
\(R = 80 - 20\)
\(R = 60\) marks

To compute the relative range of marks:
Relative range \( = \frac{R}{x_H + x_L}\)
\( = \frac{60}{80 + 20}\)
\( = \frac{60}{100}\)
\( = 0.60\)
In simple words: For grouped data, the range is found by subtracting the lower boundary of the first class from the upper boundary of the last class. The relative range expresses this spread as a fraction of the sum of these extreme boundaries.

🎯 Exam Tip: When dealing with grouped frequency distributions, always use the class boundaries (lower limit of the first class and upper limit of the last class) to determine \(x_L\) and \(x_H\).

Question 4. The frequency distribution of daily income (in thousand of 80 shops of an area) is as follows. Find the absolute and the relative measure of range of daily income from it.

Daily income (thousand Rs.)	5-9	10-14	15-19	20-24	25-29	30-34
No. of shops	11	20	17	13	12	7

Answer:
From the provided data, the upper limit of the last class is \(x_H = 34\) thousand Rs., and the lower limit of the initial class is \(x_L = 5\) thousand Rs.

To calculate the absolute measure of range:
Range \(R = x_H - x_L\)
\(R = 34 - 5\)
\(R = 29\) thousand Rs.

To calculate the relative measure of range (Coefficient of Range):
Coefficient of range \( = \frac{x_H - x_L}{x_H + x_L}\)
\( = \frac{R}{x_H + x_L}\)
\( = \frac{29}{34 + 5}\)
\( = \frac{29}{39}\)
\( = 0.7436\) (approximately)
In simple words: The absolute range indicates the total spread of income values from the lowest to the highest, while the relative range gives a dimensionless measure of this spread compared to the sum of the extreme incomes.

🎯 Exam Tip: Always double-check the class intervals for continuity and correct identification of the true upper and lower boundaries for range calculations in grouped data.